When my oldest was 3, the only thing we used was Sonlight’s P3/4 curriculum, which is mostly just a collection of great picture books and a little guide with some activities you could do along with the books. Very straightforward and simple. We didn’t start anything else until age 4, really. With my second, I started doing Confessions of a Homeschooler’s “Letter of the Week” (I specify because there are several different places that put out such programs). I didn’t do every activity, just a selection, but they’re fun and the kids like them.
It seems like the later kids want more school earlier, though. My current 4-year-old always wants more, more, more. One thing I like about the Letter of the Week activities is that I can set her up with an activity and then she can work on it independently for at least a couple of minutes, which lets me work with her sisters as they need it. She also has her own math and some Kumon books and such, but they require my assistance and so it can be harder to prioritize those and get them done regularly, even though she so desperately wants to “do school” like her big sisters.
Something else that I’m finding works well for setting her up with an activity and then letting her work on it independently is the “What Your Preschooler Needs to Know” activity books. There are two, one for ages 3-4 and one for ages 4-5. She’s doing the 4-5 one and that’s the only one I have experience with, but it’s a fantastic book, with lots of fun and varied activities. I just purchased (but haven’t received yet) the 3-4 one. Most of the reviews say that the 3-4 book is really most appropriate for older 2s or younger 3s, that it’s very simple, and that an older 3 or a 4 should just use the 4-5 book. So I’m not sure which would be appropriate for your 3-year-old, but they might be fun. The book “What Your Preschooler Needs to Know” can be done with them, but you can also just do the activity books. The regular book includes stories, science, art, math, and other things to read to your child or suggested activities to do with them.
For early learners, I also like the Kumon books, some of which are good for 3-year-olds. I like the ones on fine motor skills like cutting, folding, and pasting. I also like the ones with Alphabet Games and Number Games, as they’re fun and they really help with letter and number recognition. My kids adore these books. They’re not too expensive, either.
All About Reading, the pre-level 1, is also great at this age. It has a basic introduction to each letter, a fun craft to go with it, and it works on skills like rhyming, breaking words into parts, putting words together, recognizing the first sound in a word, things like that. I’d recommend that as a great pre-reading program for a 3-year-old, and I’d also recommend the program in general for learning to read.
Anyway, I don’t think a 3-year-old needs curriculum. I think they need great classic children’s books, activities like coloring and cutting, and a gentle introduction to the alphabet and numbers. But I think it’s fine to do a curriculum if you feel like you need more guidance, you won’t get to it otherwise, or your child is begging for “school” and you want something to give her.